These are two very different things, but they go together. A blog is simply a website that contains information or discussions in a diary format. Blogs are usually run by an individual or organization. Web 2.0 refers to the current modern format of the internet.
The internet used to be composed primarily of static web pages on sites run by companies or organizations and sometimes individuals (Web 1.0). They were mostly informational and didn’t change very often. This has changed over time. The internet that we see today (Web 2.0) is composed more of dynamic user-generated content like blogs and social media.
Blogs have been around since the 90’s, but they required knowledge of web design and a lot more work to maintain. As tools have been developed over the years, it has become fairly easy for anyone to start a blog. So, Web 2.0 is today’s internet as a whole which consists mostly of user-generated content mostly in the form of blogs.